The Shell Thron region corresponds to the main cardioid of the Mandelbrot set. Robert Devaney, among others, has studied the exponential Mandelbrot set. For example, see this paper: http://math.bu.edu/people/bob/papers/notes.pdf
The equation Robert Devaney iterates is , where lambda corresponds to the log of the base used for tetration. The parabolic cusp for the exponential Mandelbrot is , which corresponds to tetration for base , which corresponds to the main parabolic cusp of the Mandelbrot set for . The exponential Mandelbrot also has different n-cyclic domains outside the main cardioid, which continue branching in a similar manner to the Mandelbrot set. I wonder whether or not there are baby exponential Mandelbrot copies?

Other resources are the Mandel program by Wolf Jung, which supports exponential iterations under its transcendental mappings. The program also has excellent educational tutorials under its help menu, including discussions of quasi conformal mappings! http://www.mndynamics.com/
- Sheldon